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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E26</font></i>: 
 <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel14.html#V1" target="_self">jointly_Scott-continuous</a>
 <b>and </b><br/><a NAME="E3:35"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="waybel_6.html#V5">prime</a>
 <b>and </b><br/><a NAME="E4:35"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E5:35"/><i><font color="Green">E29</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p2">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#D12">WAYBEL11:def 12</a></i>;<br/>
<a NAME="E6:35"/><i><font color="Green">E30</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E7:35"/><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_6.html#K14">ConvergenceSpace</a> <span class="p1">(<span class="default"><a href="waybel11.html#K2">Scott-Convergence</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel11.html#T32">WAYBEL11:32</a></i>;<br/>
<a NAME="E8:35"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel14.html#E5:35"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E6:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> ;<br/>
<a NAME="E9:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E8:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E10:35"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="waybel11.html#V2">closed_under_directed_sups</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="waybel_0.html#V12">lower</a> )
 
<b>by </b><i><a class="ref" href="waybel11.html#T7">WAYBEL11:7</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>5</sub></font> =  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>;<br/>

<a NAME="E11:35"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="waybel_0.html#V1">directed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:35_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  )
 <b>and </b><br/><a NAME="E2:35_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  or  not <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<a NAME="E3:35_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E4:35_1"/><b>then </b>
not <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:35_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E5:35_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E7:35_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#R1">=</a>  <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="waybel14.html#E2:35"><i><font color="Green">E26</font></i></a>, <a class="txt" href="waybel14.html#E7:35"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel14.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:35_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel14.html#E8:35"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel14.html#E7:35_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E9:35_1"/><i><font color="Green">E40</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E10:35_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:35_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E7:35_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_2.html#D5">WAYBEL_2:def 5</a></i>;<br/>
<a NAME="E12:35_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E5:35_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel14.html#E10:35_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E14:35_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E15:35_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="yellow_3.html#K6">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="waybel14.html#E8:35_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:35_1"/><i><font color="Green">E45</font></i>: 
( <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E12:35_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="waybel14.html#E14:35_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E19:35_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="yellow_3.html#K6">:]</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="waybel14.html#E15:35_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="waybel14.html#E17:35_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:35_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>17</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>and </b><br/><a NAME="E4:35_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:35_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel14.html#E9:35_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="waybel14.html#E3:35_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E8:35_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E7:35_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel14.html#E4:35_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel14.html#E6:35_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="waybel_2.html#D5">WAYBEL_2:def 5</a></i>;<br/><a NAME="E9:35_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E10:35_1_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> is <a href="waybel_0.html#V13">upper</a> &amp; <font color="Maroon">c<sub>15</sub></font> is <a href="waybel_0.html#V13">upper</a> )
 <b>by </b><i><a class="txt" href="waybel14.html#E19:35_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/><a NAME="E11:35_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E19:35_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel14.html#E4:35_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="waybel14.html#E6:35_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E12:35_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="waybel14.html#E8:35_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="waybel14.html#E9:35_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="waybel14.html#E10:35_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/><b>hence </b><a NAME="E13:35_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:35_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><a NAME="E3:35_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="waybel14.html#E2:35_1_1"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E5:35_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E19:35_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="waybel14.html#E3:35_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:35_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel14.html#E9:35_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:35_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/><a NAME="E8:35_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/><a NAME="E9:35_1_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="yellow_3.html#K7">]</a></span>
 <b>by </b><i><a class="txt" href="waybel14.html#E7:35_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_2.html#D5">WAYBEL_2:def 5</a></i>;<br/><b>hence </b><a NAME="E10:35_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="waybel14.html#E5:35_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="waybel14.html#E6:35_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:35_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E23:35_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E17:35_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E24:35_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E14:35_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_1.html#T145">FUNCT_1:145</a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E25:35_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E26:35_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E5:35"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E7:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel14.html#E19:35_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E27:35_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E1:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel14.html#E3:35"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel14.html#E5:35"><i><font color="Green">E29</font></i></a>, <a class="txt" href="waybel14.html#E6:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="waybel14.html#E22:35_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="waybel14.html#E25:35_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="waybel14.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E28:35_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="waybel14.html#E17:35_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="waybel14.html#E19:35_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E29:35_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E1:35_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel14.html#E27:35_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="yellow_6.html#T10">YELLOW_6:10</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:35_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E3:35_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel14.html#E1:35"><i><font color="Green">E25</font></i></a>, <a class="txt" href="waybel14.html#E4:35"><i><font color="Green">E28</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E13:35"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:35"><i><font color="Green">E33</font></i></a>, <a class="txt" href="waybel14.html#E11:35"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel11.html#D2">WAYBEL11:def 2</a></i>;<br/>
<a NAME="E14:35"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel14.html#E10:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel14.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E15:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>


</div>
